Roots and Radicals
Roots and Radicals
Radicals (also called roots) are
directly related to exponents.
Roots and Radicals
The simplest types of radicals are
square roots and cube roots.
Radicals beyond square roots and
cube roots exist, but we will not
discuss them as in depth.
Roots and Radicals
The rules for radicals that you
will learn work for all radicals –
not just square roots and cube
roots.
Roots and Radicals
The symbol used to indicate a
root is the radical symbol -
Roots and Radicals
Every radical expression has
three parts…
• Radical symbol
• Index
• Radicand
Roots and Radicals
Every radical expression has
Radical
three parts…
Index
2
49
Radicand
Roots and Radicals
The index of a radical is a whole
number greater than or equal to 2.
Roots and Radicals
The index of a square root is
always 2.
Roots and Radicals
By convention, an index of 2 is
not written since it is the smallest
possible index.
Roots and Radicals
The square root of 49 could
2
be written as 49 …
but is normally written as 49 .
Roots and Radicals
plural of index
All indices greater than 2 must be
written.
The index of a cube root is
always 3.
Roots and Radicals
The cube root of 64 is written as
3
64 .
Roots and Radicals
What does square root mean?
What does cube root mean?
Roots and Radicals
The square root of a number is another
number …
…which when multiplied by itself gives
back the original number.
Roots and Radicals
The cube root of a number is
another number …
…which when multiplied by itself
three times gives back the original
number.
Roots and Radicals
Example:
49  7
because
7  7  7  49
2
Also
49   7
because  7  7    7   49
2
Roots and Radicals
Example:
49
has two answers:
7 is called the positive or principal
square root.
-7 is called the negative square root.
Roots and Radicals
Example:
3
3
64  4
because
4  4  4  4  64
3
 64   4 because
3
 4 4 4   4   64
Roots and Radicals
What are the first 10 whole numbers
that are perfect squares?
2
2
2
2
2
2
2
2
2
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
2
Roots and Radicals
What are the first 10 whole numbers
that are perfect cubes?
3
3
3
3
3
3
3
3
3
1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10
3
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
Roots and Radicals
The Rules (Properties)
Multiplication
Division
a b 
a

b
a b
a
b
b may not be equal to 0.
Roots and Radicals
The Rules (Properties)
Multiplication
3
a b 
3
3
a b
Division
3
a
a
3

3
b
b
b may not be equal to 0.
Roots and Radicals
Examples:
Multiplication
3  3  33
 9 3
Division
96

6
96
6
 16  4